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Recurrence, Relations (RR) and Cellular Automata (CA)

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Added on  2019-10-12

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Learn about the tent map, 1D CA, Game of Life, and graph theory in this article. Find fix points, stable points, and cycles in the tent map. Understand the rules and iterations of 1D CA and Game of Life. Explore the concept of girth in graph theory.

Recurrence, Relations (RR) and Cellular Automata (CA)

   Added on 2019-10-12

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Recurrence, Relations (RR) and Cellular Automata (CA)2.Find the fix points for the tent map. Then g(x)=μmin(x,1−x),0≤x≤1,0≤μ≤2. Plot the location of the right fix point as function of a and the derivative there. When is this fix point stable? Can you find a 2-cycle in the tent map?Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic.If μ is less than 1 the point x=0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x=0 from any initial value of x.If μis1 all values of x less than or equal to 1/2 are fixed points of the system.If μ is greater than 1 the system has two fixed points, one at 0, and the other atμμ+1. Both fixed points are unstable i.e. a value of x close to either fixed point will move away from it, rather than towards it.Figure: Bifurcation diagram for the tent map. Higher density indicates increasedprobability of the x variable acquiring that value for the given value of the μparameter.Plotting the second return map (light line) for μ=1, together with the first return map (dark line) for μ=2.
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Figure: Tent MapThe 2-cycle is shown by the dashed square.From the dashed square, we see one point, x, of the 2-cycle is less than 1/2, while the other, y, is greater than 1/2. Then using equation abovexn+1=μxnifxn≤0.5xn+1=μ(1−xn)ifxn>0.5we see y=μ×x andx=μ×(1−y). Combining these, x=μ(1−μx). Solving for x givesx=μ1+μ2. Then y=μx=μ21+μ2.3.Consider the following 1D CA: A cell is black in next generation if and only if either of its neighbors, but not both, was black on the step before. What is the rule number? Do 5 iterations using one black cell as seed.This is Rule 90 as shown in the Image below.Figure: Rule 190 1D-CALet assume the initial state at time t=0 is given as:t=0⇒0,1,0,0,0
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